Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra Tervonen, T., Almeida-Dias, J., Figueira, J., Lahdelma, R., Salminen, P. SMAA-TRI: A Parameter Stability Analysis Method for ELECTRE TRI 1 st Revision: 8 July 2005 No. 6 2005 ISSN: 1645-2631 Instituto de Engenharia de Sistemas e Computadores de Coimbra INESC - Coimbra Rua Antero de Quental, 199; 3000-033 Coimbra; Portugal
Transcript
Instituto de Engenharia de Sistemas e Computadores de Coimbra
Institute of Systems Engineering and Computers
INESC - Coimbra
Tervonen, T., Almeida-Dias, J., Figueira, J.,
Lahdelma, R., Salminen, P.
SMAA-TRI: A Parameter Stability Analysis Method for ELECTRE TRI
1st Revision: 8 July 2005
No. 6 2005
ISSN: 1645-2631
Instituto de Engenharia de Sistemas e Computadores de Coimbra
INESC - Coimbra
Rua Antero de Quental, 199; 3000-033 Coimbra; Portugal
SMAA-TRI: A PARAMETER STABILITY ANALYSIS
METHOD FORELECTRE TRI
Tommi Tervonen∗† ‡, Juscelino Almeida–Dias†, Jose Figueira† ‡,Risto Lahdelma∗, Pekka Salminen§
8th July 2005
∗Department of Information Technology, University of Turku, FIN-20520 Turku, Finland, Phone: +358 2 333 8627,Fax: +358 2 333 8600. E-mails:[email protected], [email protected]
†Faculdade de Economia da Universidade de Coimbra, Av. Dias de Silva 165, 3004-512 Coimbra, Portugal, Phone:+351 239 790 500, Fax: +351 239 790 514. E-mails:[email protected], [email protected] (J. Almeida–Dias)
‡INESC–Coimbra, R. Antero de Quental 199, 3000-033 Coimbra,Portugal, Phone: +351 239 851 040, Fax: +351239 824 692
§University of Jyvaskyla, School of Business and Economics, P.O. Box 35, FIN-40014 Jyvaskyla, Finland, Phone:+358 14 260 1211, Fax: +358 14 260 3331. E-mail:[email protected]
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SMAA-TRI: A PARAMETER STABILITY ANALYSISMETHOD FORELECTRE TRI
Abstract
ELECTRE TRI is a multiple criteria decision aiding sorting method with a historyof successful real-life applications. In ELECTRE TRI, values for certain parameters,such as criteria weights, thresholds, category profiles, and lambda cutting level, haveto be provided. We propose a new method, SMAA-TRI, that is based on StochasticMulticriteria Acceptability Analysis (SMAA), for analyzing the stability of such pa-rameters. The stability analysis can be used for deriving robust conclusions. SMAA-TRI allows ELECTRE TRI to be used with imprecise, arbitrarily distributed valuesfor weights and the lambda cutting level. The method consists of analyzing throughMonte Carlo simulation finite spaces of arbitrarily distributed parameter values in or-der to describe for each action, the share of parameter values that have it assignedto different categories. We provide algorithms for the method, and demonstrate thereal-life applicability by re-analyzing a case study in thefield of risk assessment.
Partitioning a set of objects into groups (clusters, classes, or categories) is among the most re-searched areas in various disciplines. The first contributions on grouping techniques came fromstatistical and econometric fields in form of discriminant,logit and probit analyses (Berkson, 1944;Bliss, 1934; Fisher, 1936; Smith, 1947). During the last three decades, the development of group-ing techniques has been based on operations research and artificial intelligence. These method-ologies can be divided into two families: the non-parametric techniques such as neural networks,machine learning, rough sets and fuzzy sets (Dembczynski etal., 2002; Greco et al., 1999, 2001,2002; Pawlak, 1982; Pawlak and Słowinski, 1994), and the ones based on outranking relations(Belacel, 2000; Belacel and Boulassel, 2004; Perny, 1998; Roy and Bouyssou, 1993; Yu, 1992a)and particular cases of multiple attribute utility theory (Zopounidis and Doumpos, 1999, 2000).
The groups can be defineda priori or a posteriori and be ordered or not. In the case ofa
priori defined ordered groups the problem is called anordinal classification or sorting problem,and the objects are assigned tocategories based on upper and lower profiles, central objects orother norms (Doumpos and Zopounidis, 2002).
In the late seventies a trichotomic procedure based on the outranking approach for sortingproblems was proposed by Moscarola and Roy (see Moscarola, 1977; Moscarola and Roy, 1977;Roy, 1981). Several years later, in order to help decision making in a large banking company facedwith a problem of accepting or refusing credit requests, a new method with a name of ELECTREA was developed and applied in 10 sectors of activity (Figueira et al., 2005b). Based on theseearlier works, in 1992 a method called ELECTRE TRI (Yu, 1992a,b) emerged. It is one of themost successful and applied methods for MCDA sorting problems (see e.g. Damart et al., 2002;Dimitras et al., 2001; Gabrel, 1994; Georgopoulou et al., 2003; Mavrotas et al., 2003; Merad et al.,2004; Raju et al., 2000).
ELECTRE TRI requires an input of numerous parameters. Theseparameters can be dividedinto preference parameters (relative importance coefficients of criteria or weights, thresholds, andprofiles) and thetechnical parameter (lambda cutting level). The weight elicitation process ingeneral is one of the the most difficult problems in MCDA, because MCDA methods are sup-ported by mathematical models and therefore the preferences need to be expressed in mathemati-cal terms. There are numerous weight elicitation techniques proposed for ELECTRE methods, seee.g. Figueira and Roy (2002); Hokkanen and Salminen (1997);Mousseau (1995); Mousseau et al.(2001); Rogers and Bruen (1998). All these techniques produce different values for weights, andtherefore it is advisable to perform some kind of robustnessanalysis when they are applied (Roy,2002, 2005).
Important research has been made about parameter inferenceand robustness analysis forELECTRE TRI. The first advance on this topic was made by Mousseau and Słowinski (1998),who presented a method for inferring indifference and preference thresholds, profiles and weights,from assignment examples through non-linear optimization. A user-friendly software for thismethod was presented by Mousseau et al. (2000). A linear programming method for inferring theweights from assignment examples was introduced by Mousseau et al. (2001). Dias and Clımaco(1999, 2000) presented a method for deriving robust conclusions with imprecise parameter values
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that were defined with linear constraints. These works were combined into a unified frameworkby Dias et al. (2002), allowing to infer the parameters and toderive robust conclusions based onassignment examples. Nevertheless, inconsistent judgements can appear when these methods areapplied. Algorithms for solving interactively the inconsistencies in the inferred parameter val-ues were proposed by Mousseau et al. (2004, 2003). An approach for inferring category limitswas introduced by Ngo The and Mousseau (2002). The methodology was complemented by Diasand Mousseau (2004), who introduced a partial inference procedure for inferring the veto-relatedparameters.
In this paper we introduce the SMAA-TRI method that can be used for analyzing the robust-ness of ELECTRE TRI results based on parameter stability analysis. A parameter stability anal-
ysis consists of analyzing a space of feasible parameters for possible changes in the output of themethod. Stability analysis allows the model to include non-deterministic parameters and providesthe DMs with more output than parameter inference. SMAA-TRIis based on Stochastic Multicri-teria Acceptability Analysis (SMAA) (Lahdelma et al., 1998; Lahdelma and Salminen, 2001), thatis a family of decision support methods to aid decision makers (DMs) in discrete decision makingproblems. The SMAA methods for the ranking problem statement (see Durbach, 2005; Lahdelmaet al., 1998, 2003, 2005; Lahdelma and Salminen, 2001, 2005b; Tervonen et al., 2004) are basedon inverse weight space analysis that produces descriptivevalues characterizing the decision mak-ing problem. They have been applied in numerous real-life situations (see e.g. Hokkanen et al.,1998, 1999, 2000; Kangas et al., 2003; Lahdelma and Salminen, 2005a; Lahdelma et al., 2002,2001). SMAA-TRI is the first SMAA method for the sorting problem statement.
We demonstrate the application of SMAA-TRI by re-analyzinga case study in the field ofrisk assessment. To make the method readily applicable, we provide complete algorithms for themethod in the appendices.
The rest of the paper is organized as follows: a comprehensive description of ELECTRE TRIis presented in Section 2. SMAA-TRI is introduced in Section3. Section 4 contains the re-analysisof a case-study. We end the paper with conclusions and avenues for future research in Section 5.
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2 ELECTRE TRI
ELECTRE TRI was designed to assign a set of alternatives, objects or items (actions in general) topre-defined and ordered categories. Each category is characterized by a lower and an upper profile.The assignment of an action to a certain category results from the comparison of the action withthe profiles. The comparison is based on the credibility of the assertions “the action outranks thecategory profile andvice-versa”. In what follows, we will assume, without any loss of generality,that the scales of the criteria are ascending (therefore, all the criteria are to be maximized).In this paper, we will use the following notation:
• F = {g1, . . . ,g j, . . . ,gn} is the set or family ofcriteria. Let J denote the set of criterionindices.
• A = {a1, . . . ,ai, . . . ,am} is the set ofactions. Let I denote the set of action indices.
• C = {C1, . . . ,Ch, . . . ,Ck} is the set ofcategories in ascending preference order (C1 is the“worst” category). LetC denote the set of category indices.
• B = {b1, . . . ,bh, . . . ,bk−1} is the set ofprofiles. The profilebh is the upper limit of categoryCh and the lower limit of categoryCh+1, for all h ∈ B , whereB is the set of profile indices.
• w = (w1, . . . ,w j, . . . ,wn) is theweight vector modelling the preferences of DMs. For thesake of simplicity, let us assume that∑ j∈J w j = 1 (normalized weights).
• g j(ai) is theevaluation of actionai on criteriong j for all i ∈ I and j ∈ J .
• M is the evaluation matrix composed ofg j(ai) for all i ∈ I and j ∈ J .
The following comprehensive binary relations are used thatallow to compareai andbh:
• P is thestrict preference relation, that isaiPbh denotes the relation “ai is strictly preferredoverbh”.
• I is theindifference relation, that isaiIbh denotes the relation “ai is indifferent tobh”.
• Q is theweak preference relation, that isaiQbh denotes the relation “ai is weakly preferredoverbh”, which means hesitation between indifference and strict preference.
• R is theincomparability relation, that isaiRbh denotes that actionai andbh are incompara-ble.
• S is theoutranking relation, that isaiSbh denotes that “ai is at least as good asbh”.
• � is thepreference (weak and strict) relation.
When the relational operator is subscripted (for example,S j) it denotes that the relation holds withrespect to the criterion indexed by the subscript.
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The thresholds are denoted as follows:
• q j is theindifference threshold for the criteriong j. q = (q1, . . . ,qn) is the vector of indiffer-ence thresholds.
• p j is thepreference threshold for the criteriong j. p = (p1, . . . , pn) is the vector of preferencethresholds.
• v j is theveto threshold for the criteriong j. v = (v1, . . . ,vn) is the vector of veto thresholds.
These thresholds can also vary along the scale of each criterion, and in ELECTRE TRI they arealways defined on profiles and they can be interpreted as locally constant (see Roy and Bouyssou,1993). In what follows we will consider variable thresholds, i.e. q j(g j(bh)), p j(g j(bh)), andv j(g j(bh)).
The comparisons in ELECTRE TRI are based on the pseudo-criterion model. A pseudo-criterion is a functiong j associated with two threshold functionsq(g j(·)) and p(g j(·)) satisfyingthe following conditions, for alla,b ∈ A (Roy, 1996):
q(g j(b))−q(g j(a))≥ g j(a)−g j(b) (1)
p(g j(b))− p(g j(a))≥ g j(a)−g j(b), (2)
and such that, for alla,b ∈ A with g j(a) ≥ g j(b):
aI jb⇔ g j(a)≤ g j(b)+ q(g j(b)) (3)
aQ jb⇔ g j(b)+ q(g j(b)) < g j(a) ≤ p(g j(b)) (4)
aPjb⇔ g j(b)+ p(g j(b)) < g j(a). (5)
In Figure 1, an example of a profile scheme is presented graphically. The scheme consists of aset of four criteria{g1,g2,g3,g4} with ascending scales, and a set of three categories{C1,C2,C3}.The profileb2 is the upper bound forC2 and the lower bound forC3. The dashed lines represent twodifferent actions. The evaluations of actiona1 on the left does not require any kind of preferenceinformation since it fits perfectly on the worst category,C1, and therefore it should be assignedto it. But with a2 the situation is different: the assignment ofa2 depends on the definition ofparameter values of ELECTRE TRI.
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g1(ai)
g2(ai)
g3(ai)
g4(ai)
b1 b2
a1 a2
C1 C2 C3
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
Figure 1: An example of a profile scheme.
2.1 The construction of an outranking relation
The construction of an outranking relation requires the definition of credibility indices for theoutranking relationsaiSbh andbhSai. Let ρ(ai,bh) denote the credibility index of the assertionaiSbh. It is defined by using both a comprehensive concordance index, c(ai,bh), and a discordanceindex for each criteriong j ∈ F, that is,d j(ai,bh), for all j ∈ J . The definition ofρ(bh,ai) issimilar, with the exception that the thresholds in ELECTRE TRI are always computed based onthe criterion value of the profilebh. In what follows we only exemplify the computation for therelationaiSbh.
2.1.1 The comprehensive concordance index
The concordance index is computed by considering individually for each criteriong j the supportit provides for the assertionaiS jbh. The partial concordance index is a fuzzy index measuringwhether actionai is at least as good as profilebh on criteriong j. The partial concordance indicesare computed as follows, for allj ∈ J , i ∈ I , andh ∈ B :
c j(ai,bh) =
1, if g j(ai)≥ g j(bh)−q j(g j(bh)),
0, if g j(ai) < g j(bh)− p j(g j(bh)),
g j(ai)+p j
(
g j(bh))
−g j(bh)
p j
(
g j(bh))
−q j
(
g j(bh)) , otherwise.
(6)
Thereforec j(ai,bh) increases linearly from 0 to 1, wheng j(ai) increases in the range
[g j(bh)− p j
(
g j(bh))
, g j(bh)−q j
(
g j(bh))
[.
The definition of the partial concordance index is illustrated graphically in Figure 2.
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1
0
Interpolation zone aiS jbh
c j(ai,bh)
g j(ai)g j(bh)− p j(g j(bh))g j(bh)−q j(g j(bh))
g j(bh)g j(bh)+q j(g j(bh))
g j(bh)+ p j(g j(bh))
bhPjai bhQ jai bhI jai aiI jbh aiQ jbh aiPjbh
Figure 2: The partial concordance indexc j(ai,bh).
After computing the partial concordance indices, the comprehensive concordance index is calcu-lated as follows,
c(ai,bh) = ∑j∈J
w jc j(ai,bh). (7)
2.1.2 The partial discordance indices
The discordance of a criteriong j describes the veto effect that the criterion provides against theassertionaiS jbh. The discordance indices are computed separately for all criteria. A discordanceindex is also a fuzzy index, and it reaches the maximal value when criteriong j puts its veto againstthe outranking relation. It is minimal when the criteriong j is not discordant with that relation. Todefine the value of the discordance index on the intermediatezone a linear interpolation is used.The partial discordance indices are computed as follows, for all j ∈ J , i ∈ I , andh ∈ B :
d j(ai,bh) =
1, if g j(ai) < g j(bh)− v j(g j(bh))
0, if g j(ai)≥ g j(bh)− p j(g j(bh))
g j(bh)−g j(ai)−p j(g j(bh))v j(g j(bh))−p j(g j(bh))
, otherwise.
(8)
Therefored j(ai,bh) decreases linearly from 1 to 0, wheng j(ai) increases in the range
]g j(bh)− v j
(
g j(bh))
, g j(bh)− p j
(
g j(bh))
].
The definition of the discordance index is illustrated graphically in Figure 3.
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1
0
Interpolation zone
d j(ai,bh)
g j(ai)g j(bh)−v j(g j(bh))
g j(bh)− p j(g j(bh))
Figure 3: The partial discordance indexd j(ai,bh).
2.1.3 The credibility index: a fuzzy outranking relation
The outranking relation is constructed by defining the credibility of the assertionaiSbh as follows
ρ(ai,bh) =
c(ai,bh) ∏j∈V
1−d j(ai,bh)
1− c(ai,bh), if V 6= /0,
c(ai,bh), otherwise,
(9)
withV = { j ∈ J : d j(ai,bh) > c(ai,bh)}. (10)
Notice that whend j(ai,bh) = 1 for any j ∈ J , this implies thatρ(ai,bh) = 0.
2.1.4 Converting a fuzzy relation into a crisp one
After determining the credibility index, theλ-cutting level has to be defined. The cutting levelis used to transform the fuzzy outranking relation into a crisp one. It is defined as the smallestcredibility index value compatible with the assertionaiSbh:
ρ(ai,bh)≥λ⇒ aiSbh
ρ(ai,bh) <λ⇒¬aiSbh
ρ(bh,ai)≥λ⇒ bhSai
ρ(bh,ai) <λ⇒¬bhSai
(11)
Theλ should be in the range [0.5,1], and it describes the summation of the weights of the coalitionof criteria that must support the assertionaiSbh.
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2.2 The exploitation procedure
The objective of the exploitation procedure is to exploit the binary relations introduced in theprevious sections in order to assign actions to categories.
2.2.1 Comparing actions with profiles
The actionai and the profilebh can be compared by using the obtained relations. Based on differentcombinations, an actionai can be preferred to a profilebh (�) or vice-versa, they can be indifferent(I), or they can be incomparable (R). The fuzzy outranking relation can be decomposed into thesecrisp relations as follows:
1. aiIbh⇔ aiSbh∧bhSai
2. ai � bh⇔ aiSbh∧¬bhSai
3. bh � ai⇔¬aiSbh∧bhSai
4. aiRbh⇔¬aiSbh∧¬bhSai
The different possible relations are illustrated in Figure4.
aiSbh
bhSai
¬bhSai
¬aiSbh
bhSai
¬bhSai
aiIbh
ai � bh
bh � ai
aiRbh
Figure 4: Definition of�, I, andR based on the outranking relationS.
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2.2.2 Two rules for assigning actions to categories
The sorting procedure extends two well-known procedures: the conjunctive and the disjunctive(Figueira et al., 2005b). Based on these logics, there are two possible exploitation rules: thepessimistic and the optimistic. Let us consider two more “profiles”, b0 (which every action ispreferred to) andbk (which is preferred over all actions), and let∆ denote thedominance relation.The profiles must be connected with the dominance relation asfollows:
bk ∆bk−1∆ . . . ∆bh ∆ . . . ∆b1∆b0. (12)
These are two possible rules for assigning actions to categories:
The pessimistic rule: In the pessimistic rule, an actionai is successively compared withbk,bk−1, . . . , until aiSbk−1. Thenai is assigned to the best categoryCh such thataiSbh−1.
The optimistic rule: In the optimistic rule, an actionai is successively compared withb0,b1, . . . ,until bh � ai. Thenai is assigned to the worst categoryCh such thatbh � a.
3 SMAA-TRI
The fundamental idea of SMAA is to use Monte Carlo simulationfor exploring the weight space inorder to provide DMs with values characterizing the problem. The SMAA methodology has beendeveloped for discrete stochastic MCDA problems with multiple DMs. The SMAA-2 method(Lahdelma and Salminen, 2001) applies inverse weight spaceanalysis to describe for each actionwhat kind of preferences make it the most preferred one, or place it on any particular rank. InSMAA, the criteria evaluations can be generated based on arbitrary distributions, or they can besampled from an external source.
SMAA-TRI is developed for parameter stability analysis of ELECTRE TRI, and consists ofanalyzing finite spaces of arbitrarily distributed parameter values in order to describe for eachaction the share of parameter values that assign it to different categories. We analyze the stability ofweights and the cutting level, and consider the remaining parameters to have deterministic valuesfor easier comprehensibility. The method can easily be extended to consider non-deterministicvalues for thresholds.
For analyzing ELECTRE TRI, we will denote the input for ELECTRE TRI in SMAA-TRI asfollows:
1. The lambda cutting level is presented by a stochastic variable Λ with a density functionfL(Λ) defined within the valid range [0.5,1].
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2. The weights are represented by a weight distribution witha joint density functionfW (w)in the feasible weight spaceW . Total lack of preference information is represented in“Bayesian” spirit by a uniform weight distribution inW , that is, fW (w) = 1/vol(W ). Theweights are non-negative and normalized: the weight space is ann−1 dimensional simplexin n dimensional space:
W =
{
w ∈ Rn : w≥ 0 andn
∑j=1
w j = 1
}
. (13)
3. The data and the other parameters of ELECTRE TRI are represented by the setT ={M,B,q, p,v}. Recall thatM is the criteria evaluation matrix andB is the set of profiles.These components are considered to have deterministic values for the sake of simplicity.
SMAA-TRI produces category acceptability indices for all pairs of actions and categories.The category acceptability indexπh
i describes the share of possible parameter values that have anactionai assigned to categoryCh, and it is most conviniently expressed percentage-wise. Itis ageneralization of the rank acceptability index of SMAA-2 (Lahdelma and Salminen, 2001). Letus define acategorization function that evaluates as the category indexh to which an actionai isassigned by ELECTRE TRI:
h = K(i,Λ,w,T ), (14)
and a category membership function
mhi (λ,w,T ) =
{
1, if K(i,Λ,w,T ) = h,
0, otherwise,(15)
which is applied in computing the category acceptability index numerically as a multidimensionalintegral over the finite parameter spaces as
πhi =
Z 1
0fL(Λ)
Z
w∈WfW (w)mh
i (Λ,w,T )dwdΛ. (16)
The category acceptability index measures the stability ofthe assignment, and it can be interpretedas a fuzzy measure or a probability for membership in the category. Evidently, the category accept-ability indices are within the range [0,1], where 0 indicates that the action will never be assignedto the category, and 1 indicates that it will be assigned to the category with any combination offeasible parameter values. For each action, the acceptabilities for different categories sum to unity.If the parameters are stable, the category acceptability indices for each action should be 1 for onecategory, and 0 for the others. In this case the assignments are said here to be robust with respectto the imprecise parameters.
The category acceptability indices provide a measure of uncertainty for the results of the sensi-tivity and robustness analyses as they were considered in ELECTRE TRI. While traditional way to
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perform sensitivity analysis in ELECTRE TRI is to consider the extremes of what can be consid-ered possible values for the imprecise parameters (Merad etal., 2004), the category acceptabilityindices consider the whole space which can be determined with arbitrary joint probability distribu-tions. Therefore, while robustness analysis for ELECTRE TRI (Dias et al., 2002) provides a resultsuch as “depending on the parameter values, the action is assigned either to category 2 or 3”, theSMAA-TRI provides the result as “the action is assigned to category 2 with 5% of the feasibleparameter values, and to category 3 with 95% of the feasible parameter values”.
There are three advantages gained with the additional information:
1. The cognitive effort required in determining the extremes of the parameters considered inthe sensivity analysis is reduced, because the space can be determined to be, for example,uniformly distributed and thus small changes in the interval do not change the results dra-matically.
2. Quantifying the amount of parameter values that result in“unstable” assignment determinesthe risk related with imprecise parameters. This will laterbe demostrated in the re-analysisof the case study.
3. Weight elicitation techniques provide different weightvalues, and thus it seems more rele-vant to elicit the weights as imprecise values rather than deterministic ones (see Tervonenet al. (2004)).
In addition to providing parameter stability analysis, SMAA-TRI also allows ELECTRE TRIto be applied when multiple DMs with conflicting preferencesparticipate in the decision makingprocess. The method allows arbitrarily distributed weights, and therefore they can be defined,for example, as intervals containing the preferences of allDMs (Lahdelma and Salminen, 2001).In this case the results of the analysis (the category acceptability indices) can be used to findassignments accepted by majority of the DMs. Also the extremes of parameter combinations thatassign actions to certain categories can be computed simultaneously with the parameter stabilityanalysis.
The category acceptability indices are computed through Monte Carlo simulation, quite sim-ilarily as in SMAA-2. The algorithms for SMAA-2 together with analyses of complexity andrunning times have been presented by Lahdelma and Tervonen (2004). For more informationon weight generation technique and handling of preference information, we advise the reader toconsult Lahdelma and Tervonen (2004).
The ELECTRE TRI procedure is illustrated in Figure 5, and theSMAA-TRI simulationscheme in Figure 6. The algorithms for ELECTRE TRI are provided in Appendix A, and thealgorithms for SMAA-TRI in Appendix B.
Calculate the fuzzy outrankingindices for all pairs of actions
Convert the fuzzy outrankingindices into crisp ones
Transform the crisp outrankingELECTRE TRI
Output
relations into therelations
categories according to eitherthe pessimistic or the
optimistic rule
and profiles (credibility indices)
The input data (criteria,actions, criteria
evaluations, and emptyordered categories
Parameters Data
R, I,
Assign the actions into
assigned into a singlecategory
Figure 5: The ELECTRE TRI procedure.
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Update for each action the
Parameters
with deterministic
values
Generate samples for parameters
to be analyzed from their
corresponding distributions
ELECTRE TRI
hit−counter of the rank where it
was assigned by ELECTRE TRI
Last
iteration? No
Yes
Compute the category
acceptability indices
Category acceptability indices
and categories
Output
Distributions for
parameters to be
analyzed
Input
Output from ELECTRE−TRI
Input for ELECTRE−TRI
SMAA−TRI
Run
for all pairs of actions
Figure 6: The SMAA-TRI simulation scheme.
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4 Case study: experiments and results
In this case study we re-analyze the recent real-world application of ELECTRE TRI in the field ofrisk analysis. The original analysis is presented by Merad et al. (2004). The study concentrates onFrance’s Lorraine region, where iron has been mined for morethan a century. The undergroundmining tunnels have caused land subsidence, which has caused buildings to collapse. The objectof this study was to partition land into zones and assign these zones into predefined risk categoriesin order to decide which zones need constant surveillance. We will re-analyze the assignmentprocedure by using the data provided in the case study.
The assignment phase consists of 10 homogenous zones (actions),a1, . . . ,a10, that are evalu-ated in terms of 10 criteria,g1, . . . ,g10. The criteria are presented in Table 1 (adapted from Meradet al., 2004). There are 4 risk categories where the zones areto be assigned, Category 1 is forzones with highest risk and Category 4 for lowest. The risk categories are separated by the threeprofilesb1,b2,b3. Performances of the zones together with profiles and thresholds are presentedin Table 3 (adapted from Merad et al., 2004). The authors usedthe Revised Simos’ procedure byFigueira and Roy (2002) to elicit the criteria weights. These weights are presented in Table 2.
The authors of the original case study used lambda cutting level of 0.65, but also analyzedthe sensitivity of the results by altering the lambda to 0.7,0.75, 0.8, and 0.85. In the sensitivityanalysis also different profiles were applied, but the authors did not provide them in the paper. Theresults including the sensitivity analysis are presented in Table 4.
We performed stability analysis to this case study with SMAA-TRI. We chose cutting level tobe represented by a stochastic variable uniformly distributed in the range [0.65,0.85]. The feasibleweight space was defined with constraints provided in Table 5. These constraints are not probablythe best constraints possible, as quantifying the imprecision should have been done along with theoriginal case study.
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Table 1: Criteria of the case study.No. Criterion Nature Units Encoding (scale) Direction
risk increaseG1. Susceptibility of the mine to collapse
g1 Corrected mean stress Quanti. MPa k× 0.25×H1−τ +
applied on pillarsg2 Existence of fault Quali. (code) 0:no; 10: yes. +g3 Superimposition of pillars Quali. (code) 0: only one mined layer; +
10: two well superimposed layersor thick intermediate layer (≥ 7 m);40: two bad superimposed layersor thin intermediate layer (≥ 7 m).
g4 Size and regularity of Quali. (code) 0: large pillars; +pillars 10: small regular pillars;
20: small irregular pillats.g5 Sensitivity of rock to Quali. (code) 0: no sensitivity; +
flooding (depending on the 10: sensitive;rock type) 20: very sensitive;
30: highly sensitive.G2. Surface sensitivity
g6 Depth of the top mined layer Quanti. m Given on maps. Called H. -g7 Maximum expected Quanti. m Deduced from subsidence models.+
subsidence CalledAm.g8 Expected surface Quanti. mm/m εmax = 1.5× Am
H+
deformation (deduced fromsubsidence models)
g9 Zone extent Quanti. km2 Given on maps. +g10 Vulnerability of building Quali. (code) 5: commercial zones; +
10: isolated zones;20: grouped houses;30: long buildings;40: urban road;
Table 2: Weights of the case study.Weight w1 w2 w3 w4 w5 w6 w7 w8 w9 w10
SMAA-TRI was executed with 10000 Monte Carlo iterations. The resulting category accept-ability indices are presented in Table 6. Visualization of the results is important in SMAA methods,especially if there is a large amount of actions and/or criteria. Because the categories are orderedand therefore upwards inclusive, they are visualized with stacked columns in Figure 7.
The results of the re-analysis show the usefulness of SMAA-TRI. Although the stability anal-ysis results are quite different from the ones by Merad et al.(2004), SMAA-TRI provides moreinformation. For example, compare the sensitivity analysis results for Zone 5 in Table 4 and the
15
Table 3: Criteria performances, profiles, and thresholds.Zone Criterion
Table 4: Original results of the case study and sensitivity analysis.Zone Result Sensitivity Analysisa1 Category 1 Categories 1 and 2a2 Category 1 Categories 1 and 2a3 Category 2 Stablea4 Category 2 Stablea5 Category 4 Categories 3 and 4a6 Category 1 Categories 1 and 2a7 Category 3 Categories 3 and 4a8 Category 4 Categories 3 and 4a9 Category 1 Categories 1 and 2a10 Category 4 Stable
category acceptability indices for the same zone in Table 6.The original sensitivity analysis givesinformation that Zone 5 can be assigned to risk categories 3 or 4, and with this information theDMs (especially if they are risk-aware) should treat the zone as it would be assigned to risk cate-gory 3, which is of higher risk than category 4. But with the information provided by the categoryacceptability indices more informed decision can be made: regarding our imprecise and uncertaininformation about the parameters, we can quite safely (98% acceptability) place the zone in riskcategory 4.
In this re-analysis using imprecise weights provides some interesting results. The originalsensivitity analysis considered the assignment of Zone 10 stable, but by considering the weightsimprecise (±30%), the assignment of the zone is quite unstable. With only45% of the feasibleparameter values the zone is placed in risk category 4, and a quite large share of the feasible values(34%) places the zone in risk category 2. If the original casestudy would have been performedwith imprecise weight values, the actions chosen based on the assignment would probably havebeen quite different.
17
Figure 7: The category acceptability indices.
5 Conclusions and avenues for future research
Defining parameter values for ELECTRE TRI model is not an easytask. Moreover, if there aremultiple DMs with conflicting preferences, it might even be impossible to reach consensus aboutweight values. With our approach the possibility to define the model by using stochastic variables“solves” these problems: the lambda cutting level can be defined with imprecise value, and theweights can be defined as intervals containing the preferences of all DMs.
In this paper we presented the SMAA-TRI method that allows ELECTRE TRI to be appliedwith stochastic values for lambda cutting level and weights. The SMAA-TRI analysis results incategory acceptability indices for all pairs of actions andcategories, and these can be used toanalyze the stability of the parameters. The indices can be used also to derive robust conclusions,or if not possible, to quantify the “amount of instability” in the results induced by the impreciseparameter values.
We presented a re-analysis of the case study in which the usefulness of SMAA-TRI wasdemonstrated. By visualizing the category acceptability indices with stacked columns the un-certainty related with each assignment decision can be presented to the DMs in a comprehensibleway. We provide the algorithms for the method as appendices,and hope that SMAA-TRI willbe applied in future by decision analysts for deriving robust conclusions when ELECTRE TRI ischosen as the sorting method.
18
Acknowledgements
We thank Denis Bouyssou for fruitful comments on this paper.The work of Tommi Ter-vonen was supported a grant from Turun Yliopistosaatio and the MONET research project(POCTI/GES/37707/2001). The work of Juscelino Almeida–Dias was supported by the grantSFRH/BM/18781/2004 (Fundacao para a Ciencia e Tecnologia, Portugal). The work of JoseFigueira was partially supported by the grant SFRH/BDP/6800/2001 (Fundacao para a Cienciae Tecnologia, Portugal) and the MONET research project (POCTI/GES/37707/2001).
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Appendices
A The ELECTRE TRI algorithm
ELECTRE TRI algorithm is presented in four sequential phases:
• Phase 1 consists of calculating the fuzzy outranking indices for all pairs(ai,bh) and(bh,ai),for all i ∈ I andh ∈ B . This procedure is presented in Algorithm 1.
• In Phase 2, the fuzzy outranking relation is converted into acrisp one as defined in Section2.1.4. This procedure is presented in Algorithm 2.
• After this, in Phase 3, the crisp outranking relations are converted into relationsR, I, and�.This procedure is similar to the scheme presented in Figure 4.
• In Phase 4 the actions are assigned into categories according to the pessimistic or the opti-mistic rule. The pessimistic rule is presented in Algorithm3, and the optimistic in Algorithm4.
The input of the algorithm consists of the basic data:
• A, the set of actions,
• F, the set of criteria,
• g j(ai), ∀i ∈ I , j ∈ J , the criteria evaluations,
• C, the set of categories,Ch = ∅, ∀h ∈ C ,
the preference parameters:
• B, the set of category profiles,
• w, the weight vector,
• q j(g j(bh))∀ j ∈ J ,∀h ∈ B, the set of indifference thresholds,
• p j(g j(bh))∀ j ∈ J ,∀h ∈ B, the set of preference thresholds,
• v j(g j(bh))∀ j ∈ J ,∀h ∈ B, the set of veto thresholds,
and the technical parameterλ (the cutting level), that is defined in general by the decision analyst.The procedure assigns all actions into the pre-defined categories. The procedure leads to a partition(output) of the set of actionsA in subsets or categoriesC1, . . . ,Ck:
[
h∈C
Ch = A, and
Ch∩C` = ∅,∀h, ` ∈ C , h 6= `.
24
1: for i ∈ I do2: for h ∈ B do3: for j ∈ J do4: if g j(ai)≥ g j(bh)−q j(g j(bh)) then5: c j(ai,bh)← 16: else ifg j(ai) < g j(bh)− p j(g j(bh)) then7: c j(ai,bh)← 08: else9: c j(ai,bh)←
p j(g j(bh))+g j(ai)−g j(bh)p j(g j(bh))−q j(g j(bh))
10: end if11: if g j(ai) < g j(bh)− v j(g j(bh)) then12: d j(ai,bh)← 113: else ifg j(ai)≥ g j(bh)− p j(g j(bh)) then14: d j(ai,bh)← 015: else16: d j(ai,bh)←
g j(bh)−g j(ai)−p j(g j(bh))v j(g j(bh))−p j(g j(bh))
17: end if18: end for19: c(ai,bh)← ∑ j∈J w jc j(ai,bh)20: V ←{ j ∈ J : d j(ai,bh) > c(ai,bh)}21: if V 6= /0 then
22: ρ(ai,bh)← c(ai,bh) ∏j∈V
1−d j(ai,bh)
1− c(ai,bh)
23: else24: ρ(ai,bh)← c(ai,bh)25: end if26: end for27: end for
1: for i ∈ I do2: for h ∈ B do3: if ρ(a,bh)≥ λ then4: aiSbh
5: else6: ¬aiSbh
7: end if8: if ρ(bh,ai)≥ λ then9: bhSai
10: else11: ¬bhSai
12: end if13: end for14: end forAlgorithm 2: ELECTRE TRI, Phase 2: Converting the fuzzy outranking relation into a crisp one.
1: for i ∈ I do2: h← k
3: repeat4: h← h−15: until aiSbh
6: Ch+1←Ch+1∪{ai}7: end forAlgorithm 3: ELECTRE TRI, Phase 4, the pessimistic rule: Assigns all actions into categories.
1: for i ∈ I do2: h← 03: repeat4: h← h+15: until bh � ai
6: Ch←Ch∪{ai}7: end forAlgorithm 4: ELECTRE TRI, Phase 4, the optimistic rule: Assigns all actions into categories.
26
B The SMAA-TRI algorithm
The following symbols are used in the SMAA-TRI algorithm:c A vector of category indices.K The number of Monte Carlo simulations.
And the following functions and subroutines:RANDW() Function returning a random weight vector from weight
distribution fW .RANDλ() Function returning a value from lambda cutting level dis-
tribution fλ.ELECTRE T RI(w,λ,T ) Execution of ELECTRE TRI. Returns a vector, where in
position i is the category index in which actionai is as-signed with the given parameter values. Parameters are thefollowing: w is the weight vector,λ is the cutting level, andT is the vector of parameters that have deterministic valuesthrough the simulation (see Section 2).
The procedure is presented in Algorithm 5.
1: for k← 1 to K do2: w← RANDW()3: λ← RANDλ()4: c← ELECTRE T RI(w,λ,T )5: for i ∈ I do6: πci
i ← πci
i +17: end for8: end for9: for i ∈ I do
10: for h ∈ C do11: πh
i ← πhi /K
12: end for13: end forAlgorithm 5: The SMAA-TRI algorithm: Monte Carlo simulation to compute the category ac-ceptability indicesπh
